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Research Interests: Noncommutative ring theory, Artin-Schelter regular algebras, Iterated Ore extensions


An Artin-Schelter regular algebra, A, is a finitely generated k-algebra (k a field) which is 𝐍-graded and connected (A0 = k) and which satisfies:
  1. A has finite right global dimension.
    • Right global dimension equals the supremum of the lengths of projective resolutions of all finitely graded right modules of A.
    • Equivalently, right global dimension is equal to the length of the projective resolution of the trivial A-module k.
  2. A has finite Gelfand-Kirillov dimension.
    • GK dimension equals [lim sup logn(dimkAn)]+1.
    • Equivalently, GK dimension is equal to the number of times that 1 appears as a root in p(x) where the Hilbert series is h(x) = 1⁄p(x).
  3. ExtiA(kA,AA) is 0 unless i = d (the global dimension) in which case it equals A k(l).
A good introduction to the subject can be found in the second chapter of my advisor Daniel Rogalski's notes.

An Ore extension, R[x,σ,δ], is a noncommutative polynomial ring R[x] with a new multiplication satisfying xr = σ(r)x+δ(r) where σ is an endomorphism and δ a σ-derivation, i.e. δ(r1r2) = σ(r1)δ(r2)+δ(r1)r2.

An Iterated Ore extension, R[x1,σ1,δ1]···[xn,σn,δn], is an Ore extension where σj and δj are an endomorphism and a derivation of R[x1,σ1,δ1]···[xj-1,σj-1,δj-1], respectively. If the variables are assigned nonzero degrees and the relations xjxi = σj(xi)xj+δj(xi) are also homogeneous, and the σj are all automorphisms then the iterated Ore extension is automatically Artin-Schelter regular.

I study graded iterated Ore extensions that are generated in degree 1, have injective σ's, and (global) dimension 5. I've written a paper that provides an example of a dimension 5 Ore extension with 2 generators that was not previously known to exist. I also classify all the possible degrees of relations that define an Ore extension with 3 or 4 generators.
An early draft of my paper, "A classification of relation types of Ore extensions of dimension 5," has been posted to the ArXiv.
The results rely on some computations done in mathematica. Here is a version of the code used for the computations.
Here is a more extensive version of code used for the computations for the paper and my dissertation.

I am currently generalizing these results to bigraded dimension 5 algebras that are not necessarily Ore extensions and expect to find that one of the relation types I identify in the paper cannot be realized by any bigraded algebra. This would be the first example we have of a type of AS-algebra that we know cannot be realized by any bigraded algebra.

My research statement summarizes what I have done, am currently working on, and plan to think about in the coming years. It includes projects that would be appropriate for graduate and undergraduate students as well as questions that would allow me to expand my research into related fields.